Question

Let $A$ be a square matrix, all of whose entries are integers. Then, which one of the following is true?

• A. If $det\left(A\right)=\pm 1$, then $A^{-1}$ need not exist
• B. If $det\left(A\right)=\pm 1$, then $A^{-1}$ exists but all its entries are not necessarily integers
• C. If $det\left(A\right)\ne \pm 1$, then $A^{-1}$ exists and all its entries are non-integers
• D. If $det\left(A\right)=\pm 1$, then $A^{-1}$ exists and all its entries are integers

Answer 1

The correct option is D If $det\left(A\right)=\pm 1$, then $A^{-1}$ exists and all its entries are integers

Suppose $det\left(A\right)=\pm 1$, then $A^{-1}$ exists and $A^{-1}=\frac{1}{det\left(A\right)}\left(adjA\right)=\pm adj\left(A\right)$
Since, all entries in $adj\left(A\right)$ are integers.
$\therefore A^{-1}$ has integers entries.

Hence, option D is correct.